The instanton contributions to Yang-Mills theory on the torus: localization, Wilson loops and the perturbative expansion

TL;DR

This paper derives exact formulas for instanton contributions to the partition function and Wilson loops in U(N) Yang-Mills theory on a torus, linking localization, classical solutions, and perturbative series.

Contribution

It provides explicit localization formulas for instanton effects and connects the zero-instanton sector to perturbative expansions, confirming the WML gauge approach.

Findings

Exact partition function expressed as localized contributions.

Wilson loop averages explicitly computed for N=2,3.

Zero-instanton sector reproduces perturbative series on the plane.

Abstract

The instanton contributions to the partition function and to homologically trivial Wilson loops for a U(N) Yang-Mills theory on a torus $T^2$ are analyzed. An exact expression for the partition function is obtained as a sum of contributions localized around the classical solutions of Yang-Mills equations, that appear according to the general classification of Atiyah and Bott. Explicit expressions for the exact Wilson loop averages are obtained when N=2, N=3. For general $N$ the contribution of the zero-instanton sector has been carefully derived in the decompactification limit, reproducing the sum of the perturbative series on the plane, in which the light-cone gauge Yang-Mills propagator is prescribed according to Wu-Mandelstam-Leibbrandt (WML). Agreement with the results coming from $S^2$ is therefore obtained, confirming the truly perturbative nature of the WML computations.